The (circular) helicoid is the minimal surface having a (circular) helix as its boundary. It is the only
ruledminimal surface
other than the plane (Catalan 1842, do Carmo 1986). For
many years, the helicoid remained the only known example of a complete embedded minimal surface of finite topology with infinite
curvature. However, in 1992 a second example, known
as Hoffman's minimal surface and consisting
of a helicoid with a hole, was discovered (Sci. News
1992). The helicoid is the only non-rotary surface which can glide along itself (Steinhaus
1999, p. 231).

making the helicoid a minimal surface. The Gaussian
curvature can be given implicitly by

(17)

(18)

The helicoid can be continuously deformed into a catenoid
by the transformation

(19)

(20)

(21)

where corresponds to a helicoid and
to a catenoid.

If a twisted curve
(i.e., one with torsion ) rotates about a fixed axis and, at the same time, is displaced parallel to such that the speed of displacement is always proportional
to the angular velocity of rotation, then generates a generalized
helicoid.